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<p>In this paper, we prove that for any smooth separated <i>p</i>-adic formal scheme <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3696_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathfrak {X}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">X</mi> </math></EquationSource> </InlineEquation>, the category of Laurent <i>F</i>-crystals on its absolute prismatic site <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3696_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathfrak {X}}_{{{\mathbb {\Delta }}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">X</mi> <mi mathvariant="normal">Δ</mi> </msub> </math></EquationSource> </InlineEquation> is equivalent to the category of étale <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3696_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {Z}}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-local systems on the generic fiber of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3696_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathfrak {X}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">X</mi> </math></EquationSource> </InlineEquation>. We then prove an étale comparison theorem with coefficients.</p>

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Relative \((\varphi ,\Gamma )\)-modules and prismatic F-crystals

  • 期刊论文

  • 发表日期：

    2025年3月27日

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  • Yu Min,
  • Yupeng Wang

摘要

In this paper, we prove that for any smooth separated p-adic formal scheme \({\mathfrak {X}}\) X , the category of Laurent F-crystals on its absolute prismatic site \({\mathfrak {X}}_{{{\mathbb {\Delta }}}}\) X Δ is equivalent to the category of étale \({\mathbb {Z}}_p\) Z p -local systems on the generic fiber of \({\mathfrak {X}}\) X . We then prove an étale comparison theorem with coefficients.